Digital Filters Part 12 ppt

Complex Coefcient IIR Digital Filters 211 In this chapter we examine IIR (Infinite Impulse Response) digital filters only. They are more difficult to synthesize but are more efficient and selective than FIR (Finite Impulse Response) filters. In general, the choice between FIR and IIR digital filters affects both the filter design process and the implementation of the filter. FIR filters are sufficient for most filtering applications, due to their two main advantages: an exact linear phase response and permanent stability. 1.2 Complex Signals and Complex Filters – an Overview A complex signal is usually depicted by:             tjXtXAetjtAtX IR tj CC C   sincos (1) where “R” and “I” indicate real and imaginary components. The spectrum of the complex signal X(t) is in the positive frequency  C , while that of the real one X R (t) is in the frequencies  C and -  C . There are two well-known approaches to the complex representation of the signals – by inphase and quadrature components, and using the concept of analytical representation. These approaches differ in the way the imaginary part of the complex signal is formed. The first approach can be regarded as a low-frequency envelope modulation using a complex carrier signal. In the frequency domain this means linear translation of the spectrum by a step of  C . Thus, a narrowband signal with the frequency of  C can be represented as an envelope (the real part of the complex signal – X R (t)), multiplied by a complex exponent tj C e  , named cissoid (Crystal & Ehrman, 1968) or complexoid (Martin, 2003) (Fig. 1). X R (t) X(t)=X R (t)e j  c t = X R (t)[cos( C t)+jsin( C t)] e j  c t Fig. 1. Complex representation of a narrowband signal. Analytical representation is the second basic approach to displaying complex signals. The negative frequency components are simply reduced to zero and a complex signal named analytic is formed. The real signal and its Hilbert transform are respectively the real and imaginary parts of the analytic signal, which occupies half of the real signal frequency band while its real and imaginary components have the same amplitude and 90 phase-shift. Analytic signals are, for example, the multiplexed OFDM (Orthogonal Frequency Division Multiplexing) symbols in wireless communication systems. Complex signals are easily processed by complex circuits, in which complex coefficient digital filters play a special role. In contrast to real coefficient filters, their magnitude responses are not symmetric with respect to the zero frequency. A bandpass (BP) complex filter, which is arithmetically symmetric with regards to its central frequency, can be derived by linear translation with a step  of the magnitude response of a real lowpass (LP) filter (Crystal & Ehrman, 1968). This is equivalent to applying the substitution:     sincos jzezz 1j11 (2) to the real transfer function (also called real-prototype transfer function) thus obtaining the analytical expression of the complex transfer function:           zjHzHzHzH IRComplex jzz alRe 11     sincos . (3) H Complex (z) is a transfer function with complex coefficients and with the same order of N as the real prototype H Real (z), while its real and imaginary parts H R (z) and H I (z) are of doubled order 2N real coefficient transfer functions. When H Real (z) is an LP transfer function then H R (z) and H I (z) are of BP type. For a highpass (HP) real prototype transfer function we get H R (z) and H I (z), respectively of BP and bandstop (BS) types. The substitution (2) is also termed “pole rotation” because it rotates the poles of the real transfer function to an angle of  both clockwise and anti-clockwise, simultaneously doubling their number (Fig. 2).  Poles of complex filter Pole of first-order real filter Re[z] Im[z] Fig. 2. Pole rotation of a first-order real transfer function after applying the substitution (2). Starting with:       zXzHzY Complex  (4) and supposing that the quantities in (4) are complex, they can be represented by their real and imaginary parts:                   .zjHzHzH;zjXzXzX;zjYzYzY IRComplexIRIR  (5) Then the equation (4) becomes:                                   ,zXzHzXzHjzXzHzXzH zjXzXzjHzHzY IRRIIIRR IRIR   (6) Digital Filters212 and its real and imaginary parts respectively are:                     zXzHzXzHzY;zXzHzXzHzY IRRIIIIRRR  . (7) According to the equations (7), the block-diagram of a complex filter will be as shown in Fig. 3. H R (z) H I (z) H I (z) H R (z) input R X R (z) Y I (z) + + output R Y R (z) output I X I (z) input I Fig. 3. Block-diagram of a complex filter. The synthesis of a complex filter is an important procedure because its sensitivity is influenced by the derived realization. A non-canonic complex filter realization will be obtained if H R (z) and H I (z) are synthesised individually. The process of synthesising the complex filter can be better understood by examining a particular filter realization – a real LP first-order filter section (Fig. 4a) with transfer function:   1 1 1 Real 1 1      za z zH LP . (8) The complex transfer function obtained after the substitution (2) is applied to the real transfer function (8) is:       1 1 1 1 11 sincos 1 1 1 sincos1 sincos1 1 1 11              zjaza zjz zH za z zH Complex jzzLP Real . (9) The separation of its real and imaginary parts produces:           . cos21 sin1 cos21 sincos11 22 1 1 1 1 1 22 1 1 1 2 1 1 1           zaza za j zaza zaza zjHzHzH IRComplex (10) х(n) + a 1 + y(n) z -1 + - + + z -1 + z -1 x I (n) cos  x R (n) + + cos  cos cos sin  sin  -a 1 -a 1 y I (n) y R (n) Watanabe-Nishihara method Direct realization ( b ) ( c ) ( a ) a 1 a 1 + cos  + + sin  sin  z -1 + cos  z -1 + + x I (n) x R (n) y I (n) y R (n) Fig. 4. Realization of (а) real LP first-order filter section; (b) direct-form complex BP filter section; (c) complex BP filter (Watanabe-Nishihara method). The difference equation corresponding to the transfer function (9) is:                             .1sin1cos1sin1cos 1sin1cos1sin1cos 11 11                nyanyanxnxnxj nyanyanxnxnxnjyny RIRII IRIRRIR (11) Direct realization of (11) leads to the structure depicted in Fig. 4b. Obviously the realization is canonic only with respect to the delays. The direct realization of complex filters is studied in some publications (Sim, 1987) although the sensitivity is not minimized. One of the best methods for the realization of complex structures is offered by Watanabe and Nishihara (Watanabe & Nishihara, 1991). The structure of the real prototype is doubled, for the real input and output as well as for the imaginary input and output (Fig. 5). Bearing in mind that processed signals are complex, after applying the complex transformation (2) the signals after each delay unit are described as:     .cossin;sincos 11   IRIIRR AAzBAAzB (12) Applying the Watanabe-Nishihara method to the real LP first-order filter section in Fig. 4a, the complex filter shown in Fig. 4c is derived. Complex Coefcient IIR Digital Filters 213 and its real and imaginary parts respectively are:                     zXzHzXzHzY;zXzHzXzHzY IRRIIIIRRR     . (7) According to the equations (7), the block-diagram of a complex filter will be as shown in Fig. 3. H R (z) H I (z) H I (z) H R (z) input R X R (z) Y I (z) + + output R Y R (z) output I X I (z) input I Fig. 3. Block-diagram of a complex filter. The synthesis of a complex filter is an important procedure because its sensitivity is influenced by the derived realization. A non-canonic complex filter realization will be obtained if H R (z) and H I (z) are synthesised individually. The process of synthesising the complex filter can be better understood by examining a particular filter realization – a real LP first-order filter section (Fig. 4a) with transfer function:   1 1 1 Real 1 1      za z zH LP . (8) The complex transfer function obtained after the substitution (2) is applied to the real transfer function (8) is:       1 1 1 1 11 sincos 1 1 1 sincos1 sincos1 1 1 11              zjaza zjz zH za z zH Complex jzzLP Real . (9) The separation of its real and imaginary parts produces:           . cos21 sin1 cos21 sincos11 22 1 1 1 1 1 22 1 1 1 2 1 1 1           zaza za j zaza zaza zjHzHzH IRComplex (10) х(n) + a 1 + y(n) z -1 + - + + z -1 + z -1 x I (n) cos x R (n) + + cos cos cos sin sin -a 1 -a 1 y I (n) y R (n) Watanabe-Nishihara method Direct realization ( b ) ( c ) ( a ) a 1 a 1 + cos  + + sin  sin  z -1 + cos  z -1 + + x I (n) x R (n) y I (n) y R (n) Fig. 4. Realization of (а) real LP first-order filter section; (b) direct-form complex BP filter section; (c) complex BP filter (Watanabe-Nishihara method). The difference equation corresponding to the transfer function (9) is:                             .1sin1cos1sin1cos 1sin1cos1sin1cos 11 11   nyanyanxnxnxj nyanyanxnxnxnjyny RIRII IRIRRIR (11) Direct realization of (11) leads to the structure depicted in Fig. 4b. Obviously the realization is canonic only with respect to the delays. The direct realization of complex filters is studied in some publications (Sim, 1987) although the sensitivity is not minimized. One of the best methods for the realization of complex structures is offered by Watanabe and Nishihara (Watanabe & Nishihara, 1991). The structure of the real prototype is doubled, for the real input and output as well as for the imaginary input and output (Fig. 5). Bearing in mind that processed signals are complex, after applying the complex transformation (2) the signals after each delay unit are described as:     .cossin;sincos 11   IRIIRR AAzBAAzB (12) Applying the Watanabe-Nishihara method to the real LP first-order filter section in Fig. 4a, the complex filter shown in Fig. 4c is derived. Digital Filters214 X I Y I H Real (z) + + sin sin cos z -1 cos A R B R X R Y R H Real (z) z -1 B I A I z -1 A R B R X R Y R H Real (z) X I Y I H Real (z) B I A I z -1     sincos 11 jzz Fig. 5. Complex structure realized by Watanabe and Nishihara method. The Watanabe-Nishihara method is universally applicable to any real structure, the complex structure obtained being canonic with respect to the multipliers and delay units if the sin- and cosin-multipliers are not counted. Moreover, the number of identical circuit transformations performed and the number of multipliers in the real filter-prototype are the same. A special class of filters, named orthogonal complex filters, is derived (Sim, 1987) (Watanabe & Nishihara, 1991) (Nie et al., 1993), when  is exactly equal to /2 in the complex transformation (2):            2 sin 2 cos 11 jzz or jzz  . (13) These filters are used for narrowband signal processing. Obtained after the orthogonal transformation (13) is applied, the orthogonal complex transfer function H(-jz) has alternately-changing coefficients, i.e. real and imaginary. The magnitude response of an orthogonal complex filter is symmetric with respect to the central frequency  c , which is exactly 1/4 of the real filter’s sampling frequency s. 1.3 Sensitivity Considerations Digital filters are prone to problems from two main sources of error. The first is known as transfer function sensitivity with respect to coefficients and refers to the quantization of multiplier coefficients, which changes the transfer function carried out by the filter. The second source of error is roundoff noise due to finite arithmetical operations, which degrades the signal-to-noise ratio (SNR) at the digital filter output. These errors have been extensively discussed in the literature. In this chapter normalized (classical or Bode) sensitivity is used to estimate how the changes of a given multiplier coefficient  influence the magnitude response of the structure:              jH jH S jH . (14) The overall sensitivity to all multiplier coefficients is evaluated using the worst-case sensitivity         i eHeH j i j i SWS , (15) or the Schoefler sensitivity (SS), defined as WS but with quadratic addends (Proakis & Manolakis, 2006):     2     i eHeH j i j i SSS . (16) Minimization of sensitivity is a well-studied problem but the method that is most widely used by researchers is sensitivity minimization by coefficient conversion. In this chapter we use Nishihara’s coefficient conversion approach (Nishihara, 1980). The sensitivity of magnitude, phase response, group-delay etc. is a function of frequency. This has to be taken into account when different digital structures are compared to each other because the sensitivity may differ in the different frequency bands. An indirect criterion for the sensitivity of a transfer function in a particular frequency band is the pole- location density in the corresponding area of the unit circle for a given word-length. Frequency-dependent sensitivities allow different digital filter realizations to be compared to each other in a wide frequency range. For this reason, magnitude sensitivity function (14) and worst-case sensitivity (15) will mainly be considered in this work. 2. Orthogonal Complex IIR Digital Filters – Synthesis and Sensitivity Investigations 2.1 Introductory Considerations The synthesis of orthogonal complex low-sensitivity canonic first- and second-order digital filter sections allows an efficient orthogonal cascade filter to be achieved. Such a filter can be developed using the method of approximation and design given in (Stoyanov et al., 1997). The procedure is simple in the case of arithmetically symmetric BP/BS specifications and consists of the following steps: 1. Shift the specifications along the frequency axis until the zero frequency becomes central for them. 2. Apply any possible LP or HP (for BS specifications) approximation, which produces the transfer function in a factored form. 3. Select or develop low-sensitivity canonic first- and second-order LP/HP filter sections. 4. Apply the circuit transform (13) 11   jzz to obtain the orthogonal sections, which are used to form the desired orthogonal complex BP/BS cascade realization. Complex Coefcient IIR Digital Filters 215 X I Y I H Real (z) + + sin  sin cos  z -1 cos  A R B R X R Y R H Real (z) z -1 B I A I z -1 A R B R X R Y R H Real (z) X I Y I H Real (z) B I A I z -1     sincos 11 jzz Fig. 5. Complex structure realized by Watanabe and Nishihara method. The Watanabe-Nishihara method is universally applicable to any real structure, the complex structure obtained being canonic with respect to the multipliers and delay units if the sin- and cosin-multipliers are not counted. Moreover, the number of identical circuit transformations performed and the number of multipliers in the real filter-prototype are the same. A special class of filters, named orthogonal complex filters, is derived (Sim, 1987) (Watanabe & Nishihara, 1991) (Nie et al., 1993), when  is exactly equal to /2 in the complex transformation (2):            2 sin 2 cos 11 jzz or jzz   . (13) These filters are used for narrowband signal processing. Obtained after the orthogonal transformation (13) is applied, the orthogonal complex transfer function H(-jz) has alternately-changing coefficients, i.e. real and imaginary. The magnitude response of an orthogonal complex filter is symmetric with respect to the central frequency  c , which is exactly 1/4 of the real filter’s sampling frequency s. 1.3 Sensitivity Considerations Digital filters are prone to problems from two main sources of error. The first is known as transfer function sensitivity with respect to coefficients and refers to the quantization of multiplier coefficients, which changes the transfer function carried out by the filter. The second source of error is roundoff noise due to finite arithmetical operations, which degrades the signal-to-noise ratio (SNR) at the digital filter output. These errors have been extensively discussed in the literature. In this chapter normalized (classical or Bode) sensitivity is used to estimate how the changes of a given multiplier coefficient  influence the magnitude response of the structure:              jH jH S jH . (14) The overall sensitivity to all multiplier coefficients is evaluated using the worst-case sensitivity         i eHeH j i j i SWS , (15) or the Schoefler sensitivity (SS), defined as WS but with quadratic addends (Proakis & Manolakis, 2006):     2     i eHeH j i j i SSS . (16) Minimization of sensitivity is a well-studied problem but the method that is most widely used by researchers is sensitivity minimization by coefficient conversion. In this chapter we use Nishihara’s coefficient conversion approach (Nishihara, 1980). The sensitivity of magnitude, phase response, group-delay etc. is a function of frequency. This has to be taken into account when different digital structures are compared to each other because the sensitivity may differ in the different frequency bands. An indirect criterion for the sensitivity of a transfer function in a particular frequency band is the pole- location density in the corresponding area of the unit circle for a given word-length. Frequency-dependent sensitivities allow different digital filter realizations to be compared to each other in a wide frequency range. For this reason, magnitude sensitivity function (14) and worst-case sensitivity (15) will mainly be considered in this work. 2. Orthogonal Complex IIR Digital Filters – Synthesis and Sensitivity Investigations 2.1 Introductory Considerations The synthesis of orthogonal complex low-sensitivity canonic first- and second-order digital filter sections allows an efficient orthogonal cascade filter to be achieved. Such a filter can be developed using the method of approximation and design given in (Stoyanov et al., 1997). The procedure is simple in the case of arithmetically symmetric BP/BS specifications and consists of the following steps: 1. Shift the specifications along the frequency axis until the zero frequency becomes central for them. 2. Apply any possible LP or HP (for BS specifications) approximation, which produces the transfer function in a factored form. 3. Select or develop low-sensitivity canonic first- and second-order LP/HP filter sections. 4. Apply the circuit transform (13) 11   jzz to obtain the orthogonal sections, which are used to form the desired orthogonal complex BP/BS cascade realization. Digital Filters216 The procedure becomes a lot more difficult in the case of non-symmetric specifications. There are, however, methods of solving the problems but at the price of quite complicated mathematics and transformations (Takahashi et. al., 1992) (Martin, 2005). The last two steps in the above-described procedure are discussed here. Some low- sensitivity canonic first- and second-order orthogonal complex BP/BS digital filter sections are developed and their low sensitivities are experimentally demonstrated. The Watanabe-Nishihara method (Watanabe & Nishihara, 1991) is selected to develop new sections. According to this method, it is expected that the sensitivity properties of the proto- type circuit will be inherited by the orthogonal circuit obtained after the transformation. Starting from that expectation, we apply the following strategy: first select or develop very low-sensitivity LP/HP prototypes for a given pole-position and then apply the orthogonal circuit transformation to derive the orthogonal complex BP/BS digital filter sections. The selection of LP/HP first- and second-order real prototype-sections requires the following criteria to be met: - The circuits must have canonic structures; - The magnitude response must be unity for DC (in the case of LP transfer functions), likewise for fs/2 (in the case of HP transfer functions), thus providing zero magnitude sensitivity; - The sensitivity must be minimized; - Prototype sections must be free of limit cycles 2.2 Low-Sensitivity Orthogonal Complex IIR First- Order Filter Sections In order to derive a narrowband orthogonal complex BP filter, a narrowband LP real filter- prototype must be used. When the orthogonal substitution is applied to an HP real prototype, the orthogonal complex filter will have both BP and BS outputs. The most advantageous approach is to employ a universal real digital filter section, which simultaneously realizes both LP and HP transfer functions. After a comprehensive search, we selected the best two universal first-order real filter- prototype structures that meet the above-listed requirements. They are: MHNS-section (Mitra et al., 1990-a) and a low-sensitivity LS1b-structure (Fig. 6a) (Topalov & Stoyanov, 1990). When the Watanabe-Nishihara orthogonal circuit transformation is applied to the real filter- prototypes, the orthogonal complex LS1b (Fig. 6b) and MHNS filter structures are obtained (Stoyanov et al., 1996). After the orthogonal circuit transform (13) is applied to the LP real transfer function (18)   zH LP bLS 1 the resulting orthogonal complex transfer function   jzH LPbLS  1 has complex coefficients, which are alternating real and imaginary numbers. Being a complex transfer function, it can be represented by its real and imaginary parts, which are of double order and are real coefficients:       zjHzHjzH I LPbLS R LPbLSLPbLS   111 . (17) Because the real prototype section is universal, i.e. has simultaneous LP and HP outputs, the orthogonal structure has two inputs – real and imaginary, and four outputs – two real (R1 and R2) and two imaginary (I1 and I2). Thus there are eight realized transfer functions, in the form of four pairs: the two parts of each pair are identical to each other and also equal to the real and imaginary parts of the LP- and HP-based orthogonal transfer functions - (20)(23). Only (22) is of BS type, the rest are BP. The central frequency of an orthogonal filter  C is constant and is a quarter of the sampling frequency  s . input  HP output z -1 + + + + LP output (a)     1 1 211 1 1      z z zH LP bLS (18)       1 1 211 1 1 1      z z zH HP bLS (19) B R z -1 z -1 B I A I A R B R H Real (z) H Real (z) z -1 A R B I A I z -1 11   jzz input I output R1 output R2 output I2 output I1 input R + + + +  +  + z -1 z -1 + + (b)         2 2 2 1 1 1 1 1 121 121       z z HzHzH R LPbLS II LPbLS RR LPbLS ; (20)         2 2 1 1 1 1 1 1 121 21       z z HzHzH I LPbLS IR LPbLS RI LPbLS ; (21)           2 2 2 1 2 1 2 1 121 211 1       z z HzHzH R HPbLS II HPbLS RR HPbLS ; (22)         2 2 1 1 2 1 2 1 121 12       z z HzHzH I HPbLS IR HPbLS RI HPbLS ; (23) Fig. 6. LS1b orthogonal complex section derivation (Watanabe-Nishihara transformation). The same approach, when applied to the MHNS real filter-prototype section, produces the orthogonal complex MHNS structure (Stoyanov et al., 1996). Fig. 7a depicts the worst-case gain-sensitivities for the same pole positions in LS1b and MHNS universal real filter-prototypes. It is apparent that the LS1b real section shows around a hundred times lower sensitivity than the MHNS real structure in almost the entire frequency range – from 0 to  s /2. The LS1b-section realizes unity gain on both its outputs, it is canonic with respect to the multipliers and exhibits very low sensitivity in the important applications of narrowband LP and wideband HP filters. Complex Coefcient IIR Digital Filters 217 The procedure becomes a lot more difficult in the case of non-symmetric specifications. There are, however, methods of solving the problems but at the price of quite complicated mathematics and transformations (Takahashi et. al., 1992) (Martin, 2005). The last two steps in the above-described procedure are discussed here. Some low- sensitivity canonic first- and second-order orthogonal complex BP/BS digital filter sections are developed and their low sensitivities are experimentally demonstrated. The Watanabe-Nishihara method (Watanabe & Nishihara, 1991) is selected to develop new sections. According to this method, it is expected that the sensitivity properties of the proto- type circuit will be inherited by the orthogonal circuit obtained after the transformation. Starting from that expectation, we apply the following strategy: first select or develop very low-sensitivity LP/HP prototypes for a given pole-position and then apply the orthogonal circuit transformation to derive the orthogonal complex BP/BS digital filter sections. The selection of LP/HP first- and second-order real prototype-sections requires the following criteria to be met: - The circuits must have canonic structures; - The magnitude response must be unity for DC (in the case of LP transfer functions), likewise for fs/2 (in the case of HP transfer functions), thus providing zero magnitude sensitivity; - The sensitivity must be minimized; - Prototype sections must be free of limit cycles 2.2 Low-Sensitivity Orthogonal Complex IIR First- Order Filter Sections In order to derive a narrowband orthogonal complex BP filter, a narrowband LP real filter- prototype must be used. When the orthogonal substitution is applied to an HP real prototype, the orthogonal complex filter will have both BP and BS outputs. The most advantageous approach is to employ a universal real digital filter section, which simultaneously realizes both LP and HP transfer functions. After a comprehensive search, we selected the best two universal first-order real filter- prototype structures that meet the above-listed requirements. They are: MHNS-section (Mitra et al., 1990-a) and a low-sensitivity LS1b-structure (Fig. 6a) (Topalov & Stoyanov, 1990). When the Watanabe-Nishihara orthogonal circuit transformation is applied to the real filter- prototypes, the orthogonal complex LS1b (Fig. 6b) and MHNS filter structures are obtained (Stoyanov et al., 1996). After the orthogonal circuit transform (13) is applied to the LP real transfer function (18)   zH LP bLS 1 the resulting orthogonal complex transfer function   jzH LPbLS  1 has complex coefficients, which are alternating real and imaginary numbers. Being a complex transfer function, it can be represented by its real and imaginary parts, which are of double order and are real coefficients:       zjHzHjzH I LPbLS R LPbLSLPbLS   111 . (17) Because the real prototype section is universal, i.e. has simultaneous LP and HP outputs, the orthogonal structure has two inputs – real and imaginary, and four outputs – two real (R1 and R2) and two imaginary (I1 and I2). Thus there are eight realized transfer functions, in the form of four pairs: the two parts of each pair are identical to each other and also equal to the real and imaginary parts of the LP- and HP-based orthogonal transfer functions - (20)(23). Only (22) is of BS type, the rest are BP. The central frequency of an orthogonal filter  C is constant and is a quarter of the sampling frequency  s . input  HP output z -1 + + + + LP output (a)     1 1 211 1 1      z z zH LP bLS (18)       1 1 211 1 1 1      z z zH HP bLS (19) B R z -1 z -1 B I A I A R B R H Real (z) H Real (z) z -1 A R B I A I z -1 11   jzz input I output R1 output R2 output I2 output I1 input R + + + +  +  + z -1 z -1 + + (b)         2 2 2 1 1 1 1 1 121 121       z z HzHzH R LPbLS II LPbLS RR LPbLS ; (20)         2 2 1 1 1 1 1 1 121 21       z z HzHzH I LPbLS IR LPbLS RI LPbLS ; (21)           2 2 2 1 2 1 2 1 121 211 1       z z HzHzH R HPbLS II HPbLS RR HPbLS ; (22)         2 2 1 1 2 1 2 1 121 12       z z HzHzH I HPbLS IR HPbLS RI HPbLS ; (23) Fig. 6. LS1b orthogonal complex section derivation (Watanabe-Nishihara transformation). The same approach, when applied to the MHNS real filter-prototype section, produces the orthogonal complex MHNS structure (Stoyanov et al., 1996). Fig. 7a depicts the worst-case gain-sensitivities for the same pole positions in LS1b and MHNS universal real filter-prototypes. It is apparent that the LS1b real section shows around a hundred times lower sensitivity than the MHNS real structure in almost the entire frequency range – from 0 to  s /2. The LS1b-section realizes unity gain on both its outputs, it is canonic with respect to the multipliers and exhibits very low sensitivity in the important applications of narrowband LP and wideband HP filters. Digital Filters218 (a) (b) Fig. 7. Worst-case sensitivities for the LS1b and MHNS filters (a) real –prototypes (LP outputs); (b) orthogonal structures for real input – real output BP transfer functions. (a) (b) Fig. 8. Magnitude responses of the LS1b (a) and MHNS (b) orthogonal complex filter sections for different word-lengths. For the same poles (=0.99 and =0.005), the worst-case sensitivity is also investigated for the orthogonal complex structures. In Fig. 7b graphical results for the BP transfer functions   zH R LPbLS 1 and   zH R LPMHNS are presented. The sensitivity of the LP LS1b-based orthogonal section is approximately a hundred times lower over the whole frequency range. Hence, in terms of sensitivity, the orthogonal structures have the same behaviour pattern as their real filter-prototypes. Some experimental results of the magnitude responses following the quantization of  and multipliers are shown in Fig. 8. Canonic Sign-Digit Code (CSDC) is used, together with fixed point arithmetic. Narrowband BP and BS orthogonal complex filters are investigated for poles close to the unit circle (p 1,2 =j0.99). The magnitude response of the LS1b orthogonal complex filter does not deteriorate but coincides with the ideal when the word-length is 4, or even 3, bits (Fig. 8a). The MHNS orthogonal structure (Fig. 8b) is more sensitive, and its magnitude response changes significantly, for both 3-bit and 4-bit word-lengths. The pass- band expands while the attenuation in the stop-bands decreases. Hence, the low-sensitivity structure LS1b is a better choice for applications involving analytic signal processing. 2.3. Low-Sensitivity Orthogonal Complex Second-Order IIR Filter Sections In the odd-order cascade filter structures there is one first-order section, the rest being second-order. These sections may have higher sensitivity than the first-order sections and can be more seriously affected by parasitic effects - the limit cycles and quantization noises can completely disrupt the filtering process. This is why the second-order filter sections are better investigated and a large number of sections already exists. A very low-sensitivity second-order orthogonal complex filter section, named LS2, is derived and comparatively investigated (Stoyanov et. al., 1997), (Stoyanov et al., 1996). This structure, obtained after the Watanabe-Nishihara circuit transformation is applied to the LS2 real filter-prototype (Fig. 9a), is shown in Fig. 9b. All the transfer functions of the LS2 orthogonal section are of BP type except for (28), which are BS. The orthogonal complex LS2 filter section is compared with two other often-studied second- order orthogonal complex sections: DF-section (Direct Form) (Eswaran et al., 1991) and MN- section (Minimum Norm) (Nie et al., 1993). Both real filter-prototypes and orthogonal complex filters are investigated, when realizing the same poles of the transfer function, in (Stoyanov et. al., 1997), (Stoyanov et al., 1996). In Fig. 10a the worst-case gain-sensitivities for the real prototypes are depicted. The results convincingly show that the sensitivity of the LS2 real filter section is thousands of times lower than the sensitivity of the other two real sections. The LS2 section is canonic with respect to the multipliers but a higher number of adders is the price for its very low sensitivity. In Fig. 10b the worst-case gain-sensitivities of the BP transfer functions when real input and real output are used for the three orthogonal structures are shown. It is clearly seen that the LS2 orthogonal section has a tenfold lower sensitivity compared to the MN and DF orthogonal structures, while using more than three times fewer multipliers. The same results were also obtained for the other transfer functions (Stoyanov et. al., 1997). Complex Coefcient IIR Digital Filters 219 (a) (b) Fig. 7. Worst-case sensitivities for the LS1b and MHNS filters (a) real –prototypes (LP outputs); (b) orthogonal structures for real input – real output BP transfer functions. (a) (b) Fig. 8. Magnitude responses of the LS1b (a) and MHNS (b) orthogonal complex filter sections for different word-lengths. For the same poles (=0.99 and =0.005), the worst-case sensitivity is also investigated for the orthogonal complex structures. In Fig. 7b graphical results for the BP transfer functions   zH R LPbLS 1 and   zH R LPMHNS are presented. The sensitivity of the LP LS1b-based orthogonal section is approximately a hundred times lower over the whole frequency range. Hence, in terms of sensitivity, the orthogonal structures have the same behaviour pattern as their real filter-prototypes. Some experimental results of the magnitude responses following the quantization of  and multipliers are shown in Fig. 8. Canonic Sign-Digit Code (CSDC) is used, together with fixed point arithmetic. Narrowband BP and BS orthogonal complex filters are investigated for poles close to the unit circle (p 1,2 =j0.99). The magnitude response of the LS1b orthogonal complex filter does not deteriorate but coincides with the ideal when the word-length is 4, or even 3, bits (Fig. 8a). The MHNS orthogonal structure (Fig. 8b) is more sensitive, and its magnitude response changes significantly, for both 3-bit and 4-bit word-lengths. The pass- band expands while the attenuation in the stop-bands decreases. Hence, the low-sensitivity structure LS1b is a better choice for applications involving analytic signal processing. 2.3. Low-Sensitivity Orthogonal Complex Second-Order IIR Filter Sections In the odd-order cascade filter structures there is one first-order section, the rest being second-order. These sections may have higher sensitivity than the first-order sections and can be more seriously affected by parasitic effects - the limit cycles and quantization noises can completely disrupt the filtering process. This is why the second-order filter sections are better investigated and a large number of sections already exists. A very low-sensitivity second-order orthogonal complex filter section, named LS2, is derived and comparatively investigated (Stoyanov et. al., 1997), (Stoyanov et al., 1996). This structure, obtained after the Watanabe-Nishihara circuit transformation is applied to the LS2 real filter-prototype (Fig. 9a), is shown in Fig. 9b. All the transfer functions of the LS2 orthogonal section are of BP type except for (28), which are BS. The orthogonal complex LS2 filter section is compared with two other often-studied second- order orthogonal complex sections: DF-section (Direct Form) (Eswaran et al., 1991) and MN- section (Minimum Norm) (Nie et al., 1993). Both real filter-prototypes and orthogonal complex filters are investigated, when realizing the same poles of the transfer function, in (Stoyanov et. al., 1997), (Stoyanov et al., 1996). In Fig. 10a the worst-case gain-sensitivities for the real prototypes are depicted. The results convincingly show that the sensitivity of the LS2 real filter section is thousands of times lower than the sensitivity of the other two real sections. The LS2 section is canonic with respect to the multipliers but a higher number of adders is the price for its very low sensitivity. In Fig. 10b the worst-case gain-sensitivities of the BP transfer functions when real input and real output are used for the three orthogonal structures are shown. It is clearly seen that the LS2 orthogonal section has a tenfold lower sensitivity compared to the MN and DF orthogonal structures, while using more than three times fewer multipliers. The same results were also obtained for the other transfer functions (Stoyanov et. al., 1997). Digital Filters220 a b z -1 z -1 + + + + + + + + input LP output HP output 0,5 (a)       21 21 2 1221 21 5,0      zbzab zz аzH LP LS (24)           21 21 2 1221 212 5,0      zbzab zzba zH HP LS (25) B R z -1 z -1 B I A I A R B R H Real (z) H Real (z) z -1 A R B I A I z -1 11   jzz a b z -1 z -1 + + + + + + input R + + z -1 z -1 + + a b + + + + + + 0,5 0,5 input I output R2 output I1 output R1 output I2 (b)                   4 2 2 2 42 2 1 2 1 2 112221 16341 5,0        zbzbba zbzba a zHzHzH R LPLS II LPLS RR LPLS (26)                     4 2 2 2 21 2 1 2 1 2 112221 43224 5,0        zbzbba zbabaaz zHzHzH I LPLS RI LPLS RI LPLS (27)                       4 2 2 2 42 2 2 2 2 2 112221 14212 5,0        zbzbba zbzbaba zHzHzH R HPLS II HPLS RR HPLS (28)                       4 2 2 2 12 2 2 2 2 2 112221 222 5,0        zbzbba zzbababa zHzHzH I HPLS RI HPLS RI HPLS (29) Fig. 9. Orthogonal complex LS2 second-order filter section derivation. It is clear from Fig. 10a and 10b that the orthogonal structures inherit the sensitivity of their real filter-prototypes and that the shapes of the worst-case sensitivity curves are transferred from the prototypes to the orthogonal structures, becoming symmetric around the frequency  s /4. (a) (b) Fig. 10. Worst-case sensitivities for the DF, MN and LS2 filters (a) real –prototypes (LP outputs); (b) orthogonal structures for real input – real output BP transfer functions. The effect of the coefficient quantization on the magnitude responses is experimentally investigated and some of the results for the three orthogonal structures are shown in Fig. 11. (a) (b) [...]... 3 Variable Complex IIR Digital Filters 3.1 Overview Variable digital filters (VDF) with independently tunable central frequency c and bandwidth (BW) are needed for many applications such as digital audio and video processing, medical electronics, radar systems, wireless communications etc An overview of all the main approaches to the designed structures of FIR and IIR digital filters is set out in... best-known method partly solving the problem is that of Mitra, Nevuvo and Roivainen (MNR-method) (Mitra et al., 1990-b), based on parallel all-pass real or complex structures and employing truncated Taylor series expansions of the filter coefficients to calculate them after the all-pass transformations The Complex Coefficient IIR Digital Filters 223 method is good for real digital filters but in the... truncation is used The method permits the design of BP/BS filters of any even order and any possible approximation can be applied It is also free from BW limitations and from the requirement to fix some of the pass-band edge frequencies encountered in some other design methods 224 Digital Filters 3.3 High Tuning Accuracy Variable Complex Digital Filters Sections The Constantinides LP to LP spectral transformation... IIR Digital Filters 221 (a) (b) Fig 10 Worst-case sensitivities for the DF, MN and LS2 filters (a) real –prototypes (LP outputs); (b) orthogonal structures for real input – real output BP transfer functions The effect of the coefficient quantization on the magnitude responses is experimentally investigated and some of the results for the three orthogonal structures are shown in Fig 11 (a) (b) 222 Digital. .. Magnitude responses of variable BP (a) and BS (b) variable complex LS1b section for different values of  and fixed =/3 226 Digital Filters In Fig 15 the tuning of the BW of the same LS1b by changing  is demonstrated In Fig 16 the behaviour of the complex LS1b and MHNS variable digital filters in a limited wordlength is compared It is clear that the characteristics of the MHNS-based complex filter are changed... IIR Digital Filters (a) 229 (b) (c) (d) Fig 20 Magnitude responses of the variable complex BP LS2 and DF for different coefficients word-length and BW tuned (=7/10) 3.4 Design Example and Experiments To demonstrate the advantages of the proposed improved method for designing variable complex filters, a design example will be displayed (Stoyanov & Nikolova, 1999) Two eighth-order variable complex filters. .. stop-band the parallel all-pass structure shows lower sensitivity 230 Digital Filters Fig 21 Worst-case sensitivity of second-order LS2 and all-pass real digital filter sections   0  025 0,25  0,2 (а)   0  02  0,25 -0,25 (b) (c) (d) Fig 22 Magnitude responses of the variable complex BP eighth-order LS2-based and MNRbased filters – BW tuning (a,b – for /4) and central frequency tuning... sin  cos  output I1 b a + input I + sin  sin  cos  z-1 + + + output I2 0,5 Fig 17 Variable complex second-order LS2 digital filter section Fig 18 and Fig 19 show experimental results in regard to the tuning abilities of the BP (37) and BS (39) transfer functions 228 Digital Filters               (а) (b)...  It is obvious that C can be tuned without any limitations over the entire frequency range Complex Coefficient IIR Digital Filters 225 output I1 output I2 + input R  cos  + + sin  + + z + + + z-1 sin  cos  +  -1 + input I output R2 output R1 Fig 13 Variable complex LS1b digital filter structure             ... structure is shown in Fig 17 and the transfer functions that it realizes are: ˆ RR ˆ II H LS 21НЧ z   H LS12НЧ z   ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ a 1  2 a  b A z 1  2 2 a  b  2  2  b A2  C 2  z 2  2 a  b A z 3  1  b z 4  2 Dz              (37) Complex Coefficient IIR Digital Filters 227         ˆ ˆ ˆ ˆ ˆ ˆ a 4  2 a  b Cz 1  2 bACz 2  2 a  3b  4 Cz 3 ˆ RI 1 . Complex IIR Digital Filters 3.1 Overview Variable digital filters (VDF) with independently tunable central frequency  c and bandwidth (BW) are needed for many applications such as digital audio. Complex IIR Digital Filters 3.1 Overview Variable digital filters (VDF) with independently tunable central frequency  c and bandwidth (BW) are needed for many applications such as digital audio.     ,zXzHzXzHjzXzHzXzH zjXzXzjHzHzY IRRIIIRR IRIR   (6) Digital Filters2 12 and its real and imaginary parts respectively are:                     zXzHzXzHzY;zXzHzXzHzY IRRIIIIRRR  .